The culture of tipping in restaurants has been in this country forever, it was never compulsory or expected as it seems to be in the US but it's always been there no matter how much a waiter got paid, therefore a minimum or living wage has made no difference to the tipping. Most places who allow tipping don't put it through as wages and therefore staff don't get taxed on it but it's been there a long time. Many of the higher end restaurants, the ones that can easily afford to pay wages, even put a tips section on the bill, some of them have even been seen to write in a figure of 10-15% to save the customer having to work it out for themselves.
Fast food has never really had the same luxury afforded to it but many cafes and coffee shops have had tip jars on the counters for years, even fish and chip shops started putting tip jars at the registers in the 80's. As for deliveries Pizza hut had an entire ad campaign in the 80's based around the delivery boy on his bike asking for a tip.
